论文标题
明确的通用最小常数，用于组的多项式生长
Explicit universal minimal constants for polynomial growth of groups
论文作者
论文摘要
Shalom和Tao表明，在Cayley图中，多项式上部结合了单个大球的大小，这意味着基础基团具有具有指数和多项式生长程度的Nilpotent亚组，均有效地有效。第三和第四作者证明了该亚组的多项式生长程度的最佳结合，以使结果的其他部分无效。在本文中，我们证明了在多项式生长程度上的最佳结合，而没有在其他地方造成任何损失。结果，我们表明存在明确的正数$ \ varepsilon_d $，因此，在任何具有至少多项性的$ d $多项式的群体中，增长至少为$ \ varepsilon_dn^d $。我们指出了一些概率应用；特别是，我们表明，伯努利网站在Cayley图上的关键概率的差距为$ 1 $，最近被Panagiotis和Severo证明，至少是$ \ exp \ big \ big { - \ exp \ exp \ exp \ bigl \ big big \ big {17 \ exp \ feast {100 \ cd \ cd \ cd \ cd {100 \ cd {100 \ cd {100 \ cd {100 \ cd {100 \ cd {100 \ cd { 8^{100} \} \ bigr \} \ bigr \} $。
Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and degree of polynomial growth both bounded effectively. The third and fourth authors proved the optimal bound on the degree of polynomial growth of this subgroup, at the expense of making some other parts of the result ineffective. In the present paper we prove the optimal bound on the degree of polynomial growth without making any losses elsewhere. As a consequence, we show that there exist explicit positive numbers $\varepsilon_d$ such that in any group with growth at least a polynomial of degree $d$, the growth is at least $\varepsilon_dn^d$. We indicate some applications in probability; in particular, we show that the gap at $1$ for the critical probability for Bernoulli site percolation on a Cayley graph, recently proven to exist by Panagiotis and Severo, is at least $\exp\bigl\{-\exp\bigl\{17 \exp\{100 \cdot 8^{100}\}\bigr\}\bigr\}$.
